A limit theorem for selectors

Abstract

Any (measurable) function $K$ from ${\mathbb{R}}^n$ to $\mathbb{R}$ defines an operator $\mathbf{K}$ acting on random variables $X$ by $\mathbf{K}(X)=K(X_1,\dots ,X_n)$, where the $X_j$ are independent copies of $X$. The main result of this paper concerns selectors $H$, continuous functions defined in ${\mathbb{R}}^n$ and such that $H(x_1,x_2,\dots ,x_n)\in \{x_1,x_2,\dots ,x_n\}$. For each such selector $H$ (except for projections onto a single coordinate) there is a unique point ${\omega }_H$ in the interval $(0,1)$ so that for any random variable $X$ the iterates ${\mathbf{H}}^{(N)}$ acting on $X$ converge in distribution as $N\to \infty$ to the ${\omega }_H$-quantile of $X$.